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[identity profile] writinghawk.livejournal.com Wed 2014-03-12 20:36
Ok, sorry about earlier brainstorm. When I saw your 7/4 and saw that that was one of the numbers in my calculation, I was hypnotised briefly by my earlier mistake into thinking 7/4 > 9/5.

As usual when one has been staring at something too long, the truth is perfectly simple - I was over-complicating things with that max(). The two min()'s give two constraints, one of which will be too strong (overconstrained), so I should take the lower of them - simply, the min of the 4 quantities. And two of them can be ruled out: n/(p+1) < n/p, and similarly m-n/p < m-n/(p+1), so the latter quantity in each inequality can be discarded.

I.e. if m and n are coprime, m>2, and

p = floor (2n/m)

s' = min [ m-n/p, n/(p+1) ]

Then s' is the bound: s(m,n) <= s' and very often s=s'.

If m=1, s=1. If m=2 and n is odd, s=1. If m = dm', n=dn' for some integer d>1 then s(m,n) = d s(m',n').

Does this agree with what you calculated?
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